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Intermediate mathematics
See also: Category:Foundations A generalization (or generalisation) is the formulation of general concepts from specific instances by abstracting common properties. Generalization is the process of identifying the parts of a whole, as belonging to the whole. The parts, completely unrelated may be brought together as a group by establishing a common relation between them.Wikipedia:Generalization Foreword Mathematical notation can be extremely intimidating. Wikipedia is full of articles with page after page of indecipherable text. At first glance this article might appear to be the same. I want to assure the reader that every effort has been made to simplify everything as much as possible and to provide all relevant information or, at least, to make such information easy to find. Numbers The basis of all of mathematics is the "Next" function (see Graph theory). Next(0)=1, Next(1)=2, Next(2)=3, Next(3)=4...This defines the whole numbers. Addition is defined as repeatedly calling the Next function, and its inverse is subtraction. But this leads to the ability to write equations like 1-3=x for which there is no answer among the whole numbers. To provide an answer mathematicians generalize the set of whole numbers to the set of all integers which includes negative integers. Multiplication is defined as repeated addition, and its inverse is division. But this leads to equations like 3/2=x for which there is no answer among integers. So mathematicians generalize the set of integers to the set of rational numbers. Exponentiation is defined as repeated multiplication, and its inverse is the nth root. But this leads to 2 problems: :Equations like \sqrt{2}=x for which there is no answer among rational numbers so mathematicians generalize the set of rational numbers to the set of real numbers. :Equations like \sqrt{-1}=x for which there is no answer among real numbers so mathematicians generalize the set of real numbers to the set of complex numbers by defining i = \sqrt{-1}. ::Hypercomplex numbers like quaternions are one way to generalize complex numbers to some (but not all) higher dimensions. Imaginary numbers often occur in equations involving change with respect to time. If friction is resistance to motion then imaginary friction would be resistance to change of motion wrt time. (In other words, imaginary friction would be mass.) In fact, in the equation for the Spacetime interval, time itself is an imaginary quantity. Complex numbers can be used to represent and actually perform rotations but only in 2 dimensions. Tensors, on the other hand, can be used to represent and perform rotations (and other linear transformations) in any number of dimensions. Rotations in n dimensions are called SO(n). See Graphical explanation of Tensor components. A tensor is a multivector. Understanding how tensors work leads to geometric algebra. Clifford algebra is a generalization of geometric algebra. An equation written in geometric algebra is much more intuitive and therefore easier to understand than when written in matrix form. Geometry :'' {\mathbf e_1} , {\mathbf e_2} , {\mathbf e_3} are orthogonal unit basis vectors). : {\mathbf u} , {\mathbf v} , {\mathbf x} are arbitrary vectors.'' ::'' {\mathbf u} = {\mathbf u_1}{\mathbf e_1} + {\mathbf u_2}{\mathbf e_2} + {\mathbf u_3}{\mathbf e_3} '' ::'' {\mathbf v} = {\mathbf v_1}{\mathbf e_1} + {\mathbf v_2}{\mathbf e_2} + {\mathbf v_3}{\mathbf e_3} '' ::'' {\mathbf x} = {\mathbf x_1}{\mathbf e_1} + {\mathbf x_2}{\mathbf e_2} + {\mathbf x_3}{\mathbf e_3} '' The one dimensional number line can be generalized to a multidimensional Coordinate system thereby creating multidimensional math (i.e. geometry). A vector space is a coordinate space with vector addition and scalar multiplication (multiplication of a vector and a scalar belonging to a field). :A module generalizes a vector space by allowing multiplication of a vector and a scalar belonging to a ring. Coordinate systems define the length of vectors parallel to one of the axes but leave all other lengths undefined. This concept of "length" which only works for certain vectors is generalized as the "norm" which works for all vectors. The norm of vector v is \|\mathbf{v}\|. In Euclidean space \|\mathbf{v}\|^2 = v_1^2 + v_2^2 + v_3^2. See Pythagorean theorem. A topological space is a generalization of a metric space which is a generalization of a normed vector space. A manifold (a type of topological space) is a generalization of Euclidean space. :Informally, a tangent bundle \mathbf{TM} on a differentiable manifold \mathbf{M} (blue circle in image to the right) is obtained by joining all the tangent spaces (red lines) together in a smooth and non-overlapping manner (red cylinder).Wikipedia:Tangent bundle The tangent bundle always has twice as many dimensions as the manifold. ::A fiber bundle is a generalization of a vector bundle which is a generalization of a tangent bundle. Multiplication can be generalized to allow for Multiplication of vectors in 5 different ways: Dot product (a Scalar): \mathbf{u}\bullet\mathbf{v} = \|\mathbf{u}\|\ \|\mathbf{v}\|\cos(\theta) = u_1 v_1 + u_2 v_2 + u_3 v_3 : \mathbf{u}\bullet\mathbf{v} = \begin{bmatrix}u_1 \mathbf{e_1} \\ u_2 \mathbf{e_2} \\ u_3 \mathbf{e_3} \end{bmatrix} \begin{bmatrix}v_1 \mathbf{e_1} & v_2 \mathbf{e_2} & v_3 \mathbf{e_3} \end{bmatrix} = \begin{bmatrix}u_1 v_1 \mathbf{e_1}\mathbf{e_1} + u_2 v_2 \mathbf{e_2}\mathbf{e_2} + u_3 v_3 \mathbf{e_3}\mathbf{e_3} \end{bmatrix} :Strangely, only parallel components multiply. In Euclidean space \|\mathbf{v}\|^2 = \mathbf{v}\bullet\mathbf{v}. The dot product of a rank n tensor and a rank m tensor results in a rank n-m tensor. ::The dot product can be generalized to the bilinear form \beta(\mathbf{u,v}) = scalar and its associated quadratic form Q(\mathbf{x}) = \beta(\mathbf{x,x}). Two vectors are orthogonal if \beta(\mathbf{u,v}) = 0. :::The bilinear form can be further generalized to the Sesquilinear form (an inner product is a sesquilinear form). ::::A Hilbert space is an inner product space that is also a Complete metric space. Outer product (a tensor): \mathbf{u} \otimes \mathbf{v}. :As one would expect, every component of one vector multipies with every component of the other vector. : \begin{align}\mathbf{u} \otimes \mathbf{v} = \begin{bmatrix}u_1 \mathbf{e_1} \\ u_2 \mathbf{e_2} \\ u_3 \mathbf{e_3} \end{bmatrix} \begin{bmatrix}v_1 \mathbf{e_1} & v_2 \mathbf{e_2} & v_3 \mathbf{e_3} \end{bmatrix} = \begin{bmatrix}u_1 v_1 \mathbf{e_1} \mathbf{e_1} & u_1 v_2 \mathbf{e_1} \mathbf{e_2} & u_1 v_3 \mathbf{e_1} \mathbf{e_3} \\ u_2 v_1 \mathbf{e_2} \mathbf{e_1} & u_2 v_2 \mathbf{e_2} \mathbf{e_2} & u_2 v_3 \mathbf{e_2} \mathbf{e_3} \\ u_3 v_1 \mathbf{e_3} \mathbf{e_1} & u_3 v_2 \mathbf{e_3} \mathbf{e_2} & u_3 v_3 \mathbf{e_3} \mathbf{e_3} \end{bmatrix} \end{align} ::The Tensor product generalizes the outer product. The tensor product of a rank n tensor and a rank m tensor results in a rank n+m tensor. Wedge product (a simple bivector): \mathbf{u} \wedge \mathbf{v} = \mathbf{u} \otimes \mathbf{v} - \mathbf{v} \otimes \mathbf{u} = \overline{\mathbf{v}} :The wedge product is also called the exterior product (sometimes mistakenly called the outer product). The term "exterior" comes from the exterior product of two vectors not being a vector. In three dimensions \mathbf{u} \wedge \mathbf{v} is a pseudovector and its dual is the cross product. \overline{\mathbf{u} \wedge \mathbf{v}} = \mathbf{u} \times \mathbf{v} : \mathbf{u} \wedge \mathbf{u} = 0 : ::The Matrix commutator generalizes the wedge product. :: A_2 = A_1A_2 - A_2A_1 Regressive product: \mathbf{u} \vee \mathbf{v} = \underline{\overline{\mathbf{u}} \wedge \overline{\mathbf{v}}} = \overline{\underline{\mathbf{u}} \wedge \underline{\mathbf{v}}} :The regressive product of two vectors is the dual of the wedge product of the duals of the two vectors. Geometric product (a multivector): \mathbf{u} \mathbf{v} = \mathbf{u} \bullet \mathbf{v} + \mathbf{u} \wedge \mathbf{v} : \mathbf{e_1} \mathbf{e_1} = \mathbf{e_{11}} = \mathbf{e_1} \bullet \mathbf{e_1} + \mathbf{e_1} \wedge \mathbf{e_1} = 1 + 0 = 1 : \mathbf{e_1} \mathbf{e_2} = \mathbf{e_{12}} = \mathbf{e_1} \bullet \mathbf{e_2} + \mathbf{e_1} \wedge \mathbf{e_2} = 0 + \mathbf{e_1} \wedge \mathbf{e_2} = \overline{\mathbf{e_2}} : ^2 Where \epsilon_{\mathbf{a}} is the signature of the vector. |} Integration The integral (antiderivative) is a generalization of multiplication. :For example: an object dropped from point r1 to point r2 will release energy but the usual equation mass \cdot gravity \bullet (r_1 - r_2) = energy cant be used if the strength of gravity is itself a function of radius. The strength of gravity at r1 would be different than it is at r2. And in fact g® = 1/r^2 (See inverse-square law.) :However, the corresponding Definite integral is easily solved: mass \cdot \int_{r_1}^{r_2} g® \cdot dr : The derivative is a generalization of division. The derivative of the integral of f(x) is just f(x). Partial derivatives and multiple integrals generalize derivatives and integrals to multiple dimensions. The partial derivative with respect to one variable \frac{\part f(x,y)}{\part x} is found by simply treating all other variables as though they were constants. Multiple integrals are found the same way. : The Lie derivative generalizes the Lie bracket which generalizes the wedge product which is a generalization of the cross product which only works in 3 dimensions. The cross product is neither commutative nor associative and therefore doesnt form a field or even a ring (see below). Instead it forms a Lie algebra (See Infinitesimal transformation) which is a local or linearized version of a Lie group. A Lie group is a group that is also a differentiable manifold. Generalization of addition and multiplication :Main article: Algebraic structure Addition and multiplication can be generalized in so many ways that mathematicians were forced to create categories to organize them. Category theory :See also: Higher category theory A category is an algebraic structure similar to a group but without requiring inverse or closure properties Categories consist of: :Objects (usually Sets) :Morphisms (usually functions or maps) associated with: ::one source object (domain) ::one target object (codomain) a morphism is represented by an arrow: : f(x)=y is written f : x \to y : g(a)=b is written g : a \to b Composition of morphisms: : g(f(x)) is written g \circ f A homomorphism is a map from one set to another of the same type which preserves the operations of the algebraic structure: : f(x \bullet y) = f(x) \bullet f(y) ::A Functor is a homomorphism with a domain in one category and a codomain in another. An endomorphism is a map from a set into itself. An isomorphism is invertible. An automorphism is an invertible map from a set into itself. A Multicategory has morphisms with more than one source object. A Multilinear map f(v_1,\ldots,v_n) = W : : f\colon V_1 \times \cdots \times V_n \to W\text{,} has a corresponding Linear map: F(v_1\otimes \cdots \otimes v_n) = W : : F\colon V_1 \otimes \cdots \otimes V_n \to W\text{,} Dot product might not be as strange as it first appears :See Hodge dual Let \mathbf{v} be a covector (a pseudovector) that is the orthogonal complement of vector \mathbf{v}. (A covector would be a tensor of rank -1.) \mathbf{u} \bullet \mathbf{v} = \mathbf{u} \wedge \overline{\mathbf{v}}I = \begin{bmatrix} u_1 \mathbf{e_1} \\ u_2 \mathbf{e_2} \\ u_3 \mathbf{e_3} \end{bmatrix} \wedge \begin{bmatrix} v_1 \mathbf{e_2 e_3} & v_2 \mathbf{e_3 e_1} & v_3 \mathbf{e_1 e_2} \end{bmatrix} I Therefore: \mathbf{u} \bullet \mathbf{v} = \begin{bmatrix} u_1 v_1 \mathbf{e_1 \wedge e_2 e_3} & u_1 v_2 \mathbf{e_1 \wedge e_3 e_1} & u_1 v_3 \mathbf{e_1 \wedge e_1 e_2} \\ u_2 v_1 \mathbf{e_2 \wedge e_2 e_3} & u_2 v_2 \mathbf{e_2 \wedge e_3 e_1} & u_2 v_3 \mathbf{e_2 \wedge e_1 e_2} \\ u_3 v_1 \mathbf{e_3 \wedge e_2 e_3} & u_3 v_2 \mathbf{e_3 \wedge e_3 e_1} & u_3 v_3 \mathbf{e_3 \wedge e_1 e_2} \end{bmatrix} I Which reduces to: \mathbf{u} \bullet \mathbf{v} = \begin{bmatrix} u_1 v_1 \mathbf{e_{123}} & 0 & 0 \\ 0 & u_2 v_2 \mathbf{e_{123}} & 0 \\ 0 & 0 & u_3 v_3 \mathbf{e_{123}} \end{bmatrix} I = (u_1 v_1 + u_2 v_2 + u_3 v_3) Because a trivector in 3 dimensions is a pseudoscalar: : (\mathbf{e_{123}})I = 1 ::Trivector with unit volume. : (\mathbf{e_{223}})I = 0 ::Trivector with zero volume (since its 2 dimensional). So, just like the cross product, the dot product is the dual of the wedge product. But whereas the cross product is between two vectors the dot product is between a vector and a covector. The Regressive product is between 2 covectors. References Category:Foundations